some time that he had actually found a solution, but he afterward discovered his own error. His apparent success won for him the life-long friendship and support of his countryman, Hansteens.

Abel was not the first who attempted to prove that the general quintic cannot be solved by the extraction of roots. About a quarter of a century earlier Paolo Ruffini did much to develop methods which were of sufficient power to prove this fundamental fact. In particular, he gave a number of theorems on groups of substitutions, as was noted above. The difficulties which the general solution of the quintic presented have thus become a source of great riches for the later development of mathematics. Besides Ruffini, some of the most eminent among those who started these developments are: Tschirnhaus, Euler, Lagrange, Gauss, Galois, and Hermite.

The work of Galois (1811-32) was especially fundamental as regards the establishment of more definite relations between the theory of equations and the theory of substitution groups, by proving that every equation belongs to a certain substitution group, and that the properties of this group give definite information as to the solvability by radicals of the equations belonging to the group. The important theorem that two rational functions of the roots of any equation may be expressed rationally in terms of each other, in the domain of rationality of the coefficients of the equations, had been proved earlier by Lagrange. For an introduction to the elegant theory of equations based upon these theorems we may refer the reader to the following works: Dickson, Introduction to the Theory of Algebraic Equations, 1903; Cajori, An Introduction to the Modern Theory of Equations, 1904; Mathews, Algebraic Equations, 1907.

IV.

EQUATIONS WITH ONE UNKNOWN AND WITH
NUMERICAL COEFFICIENTS

18. General statement. Although numerical algebraic equations have a prehistoric origin, the arithmetical epigrams of the Greek Anthology, among other things, support the assumption that they resulted from puzzles and word-equations. The fully developed equations represent highways of exact thought without by-ways, and the coefficients determine the possible destinations of these highways. The ancient problems of duplicating a cube and trisecting an angle, among many others, directed attention to the need of such highways, but their construction for coefficients, which may be regarded as arbitrary, presented great difficulties. Even in the case of the cubic with three real roots (casus irreducibilis) Cardan's formula represents the real root in the form of the sum of two imaginary expressions, and it has been proved that it is impossible in this case to represent the roots of the cubic in a real form by means of radicals.* On the other hand, the great French algebraist, Vieta (1540-1603), showed how the real values of the three roots may be obtained by means of trigonometry.

From the preceding paragraph it results that the solution. of numerical equations of a given degree may present difficulties even after a formula for the roots of the general equation of this degree is known. These difficulties, combined with those of finding such general formulas, directed attention. to special methods of solution in case the coefficients are numbers. It is of especial importance to observe that for many applications of algebra only approximate values of the real roots are needed. This need has led to a vast literature.

* Cf. Encyklopädie der Mathematischen Wissenschaften, Vol. I, p. 518. The French edition of this work, to which we have already referred, treats many subjects more completely than the German. This is especially true as regards algebra and arithmetic. Neither of these editions is completely published, but the German is considerably further advanced than the French. They constitute at present the most important mathematical works of refer

ence.

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where the coeficients of the rational integral functions (r) andyz are real numbers. After multiplying both members of this equation by the conjugate value, cz-igr), we obtain a new rational integral function of r, which involves all the roots of ƒ z =0, but has only real coefficients. Hence it results that if we can find all the roots of every rational integral function of z with real coefficients we can also find those of such a function with complex coefficients. It is also important to observe from the given form of ƒ that any real root of f'r,=0 is a common root of ÷ `=0,2)=0, and hence it is a root of the highest common factor of o ̧r) and (r). In view of these facts and for the sake of brevity and perspicuity we shall assume throughout the rest of the present section that all the coefficients of ƒ r) are real numbers.

19. Multiple roots. If f.r) is divisible by (r−r)a but not by (-rja+1, r is said to occur exactly a times as a root of the equation f(x)=0; sometimes it is also called such a root or a zero of f(x). When a>1, r is called a multiple root of f(x)-0, or multiple zero of f(x). To determine the multiple roots of f(x)=0 it is convenient to use the well-known property

that any root which occurs exactly a times in f(x)=0 must occur exactly a -1 times as a root of f'(x)=0, where f'(x) is the first derivative of f(x). Hence a multiple root of f(x) is also a root of the highest common factor of the two functions, f(x), f'(x). Since the first derivative of f(x) may be found by a rational process it results that f(x) is reducible in the domain of rationality of its coefficients whenever it has multiple roots, but the converse of this theorem is evidently not necessarily true.

From the preceding paragraph it results that the multiple roots of f(x)=0 may be found by means of the highest common factor of f(x) and f'(x). As the multiple roots of this highest common factor may be found in a similar manner it results that whenever f(x)=0 has no more than ẞ distinct multiple roots, all these roots may be found by rational operations and by solving equations whose degrees do exceed ẞ. In particular, if f(x)=0 has only one multiple root it may be found by rational operations. It is frequently possible to find the rational multiple roots by inspection.

Since the quotient obtained by dividing f(x) by the highest common factor of f(x) and f'(x) involves each root of f(x) once and only once, we may suppose in what follows that f(x)=0 involves no multiple root. This hypothesis will conduce to brevity of statements.

20. Sturm's theorem. This theorem (proved in 1829) furnishes the scientific foundation for every method of finding the approximate values of the unknown in an algebraic equation with real coefficients, as it gives definite information in regard to the number of real roots between two arbitrarily assigned numbers. Moreover, the proof of this theorem is not difficult, being based upon the following two elementary facts: (1) The continuity of f(x), and (2) the fact that if a is a real root of f(x)=0 and h is a sufficiently small positive number, then f(x -h) and f'(a -h) have different signs, while

* Encyklopädie der Elementar-Mathematik von Weber und Wellstein, 1906, Vol. I, p. 337.

f(a+h) and f'(a+h) must have the same sign, where f'(x) is the first derivative of f(x). A proof of these two facts is found in many elementary text-books; e.g., Burnside and Panton's Theory of Equations, Vol. I, 1899, pp. 9 and 161.

To obtain Sturm's Series we proceed exactly as in the process of finding the highest common factor of f(x) and ƒ′(x) with the single exception that the sign of each remainder is changed. In this way we obtain the following relations:

f(x)=q1(x)f'(x) −r1(x),

ƒ′(x)=q2(x)r1(x) −r2(x),

r1(x)=q3(x)r2(x) −r3(x),

rn-2(x)=qn(x)rn-1(x) −rn(X),

where rn(x) is a constant, different from zero, since ƒ(x)=0 has no multiple root. The series,

has the following properties: No two adjacent functions can vanish for the same value of x; otherwise all the succeeding functions would have to vanish for this value of x, but this is impossible since rn(x) cannot be 0. When any function vanishes the two adjacent functions must have opposite signs in order to satisfy the given equations. In finding the number of changes of sign in this series as x increases continuously from the real number a to a larger real number b we need therefore not consider the vanishing of any function except the first one. In case this vanishes a change of sign is lost, as was observed in the preceding paragraph. This proves Sturm's Theorem, which may be stated as follows:

If any two real numbers a and b be substituted for x in Sturm's Series,

f(x), f'(x), r1(x), r2(x), . rn(x),

the difference between the number of changes of sign in the series when a is substituted for x and the number when b is substituted for x is exactly the number of real roots of the equation f(x)=0 between a and b.